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Theorem ressid 13516
Description: Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Hypothesis
Ref Expression
ressid.1  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
ressid  |-  ( W  e.  X  ->  ( Ws  B )  =  W )

Proof of Theorem ressid
StepHypRef Expression
1 ssid 3359 . 2  |-  B  C_  B
2 ressid.1 . . 3  |-  B  =  ( Base `  W
)
3 fvex 5734 . . 3  |-  ( Base `  W )  e.  _V
42, 3eqeltri 2505 . 2  |-  B  e. 
_V
5 eqid 2435 . . 3  |-  ( Ws  B )  =  ( Ws  B )
65, 2ressid2 13509 . 2  |-  ( ( B  C_  B  /\  W  e.  X  /\  B  e.  _V )  ->  ( Ws  B )  =  W )
71, 4, 6mp3an13 1270 1  |-  ( W  e.  X  ->  ( Ws  B )  =  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   ` cfv 5446  (class class class)co 6073   Basecbs 13461   ↾s cress 13462
This theorem is referenced by:  submid  14743  subgid  14938  gaid2  15072  subrgid  15862  rlmsca  16263  rlmsca2  16264  pjff  16931  rlmbn  19307  ishl2  19316  evlrhm  19938  evl1sca  19942  evl1var  19944  pf1ind  19967  dchrptlem2  21041  lnmfg  27148  lmhmfgsplit  27152  pwslnmlem2  27163  dsmmfi  27172
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-ress 13468
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